Fluid Mechanics for

Chemical Engineers

Second Edition

with Microﬂuidics and CFD

Prentice Hall International Series in the

Physical and Chemical Engineering Sciences

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FLUID MECHANICS FOR

CHEMICAL ENGINEERS

Second Edition

with Microﬂuidics and CFD

JAMES O. WILKES

Department of Chemical Engineering

The University of Michigan, Ann Arbor, MI

with contributions by

STACY G. BIRMINGHAM: Non-Newtonian Flow

Mechanical Engineering Department

Grove City College, PA

BRIAN J. KIRBY: Microﬂuidics

Sibley School of Mechanical and Aerospace Engineering

Cornell University, Ithaca, NY

COMSOL (FEMLAB): Multiphysics Modeling

COMSOL, Inc., Burlington, MA

CHI-YANG CHENG: Computational Fluid Dynamics and FlowLab

Fluent, Inc., Lebanon, NH

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Library of Congress Cataloging-in-Publication Data

Wilkes, James O.

Fluid mechanics for chemical engineers, 2nd ed., with microﬂuidics

and CFD/James O. Wilkes.

p. cm.

Includes bibliographical references and index.

ISBN 0–13–148212–2 (alk. paper)

1. Chemical processes. 2. Fluid dynamics. I. Title.

TP155.7.W55 2006

660’.29–dc22

2005017816

Copyright c 2006 Pearson Education, Inc.

All rights reserved. Printed in the United States of America. This publication is protected

by copyright, and permission must be obtained from the publisher prior to any prohibited

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electronic, mechanical, photocopying, recording, or likewise. For information regarding

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ISBN 0-13-148212-2

Text printed in the United States on recycled paper at Courier Westford in Westford, Massachusetts

8th Printing

October 2012

.

Dedicated to the memory of

Terence Robert Corelli Fox

Shell Professor of Chemical Engineering

University of Cambridge, 1946–1959

This page intentionally left blank

CONTENTS

PREFACE

xv

PART I—MACROSCOPIC FLUID MECHANICS

CHAPTER 1—INTRODUCTION TO FLUID MECHANICS

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Fluid Mechanics in Chemical Engineering

General Concepts of a Fluid

Stresses, Pressure, Velocity, and the Basic Laws

Physical Properties—Density, Viscosity, and Surface Tension

Units and Systems of Units

Example 1.1—Units Conversion

Example 1.2—Mass of Air in a Room

Hydrostatics

Example 1.3—Pressure in an Oil Storage Tank

Example 1.4—Multiple Fluid Hydrostatics

Example 1.5—Pressure Variations in a Gas

Example 1.6—Hydrostatic Force on a Curved Surface

Example 1.7—Application of Archimedes’ Law

Pressure Change Caused by Rotation

Example 1.8—Overﬂow from a Spinning Container

Problems for Chapter 1

3

3

5

10

21

24

25

26

29

30

31

35

37

39

40

42

CHAPTER 2—MASS, ENERGY, AND MOMENTUM BALANCES

2.1

2.2

2.3

2.4

2.5

2.6

General Conservation Laws

Mass Balances

Example 2.1—Mass Balance for Tank Evacuation

Energy Balances

Example 2.2—Pumping n-Pentane

Bernoulli’s Equation

Applications of Bernoulli’s Equation

Example 2.3—Tank Filling

Momentum Balances

Example 2.4—Impinging Jet of Water

Example 2.5—Velocity of Wave on Water

Example 2.6—Flow Measurement by a Rotameter

vii

55

57

58

61

65

67

70

76

78

83

84

89

Contents

viii

2.7

Pressure, Velocity, and Flow Rate Measurement

Problems for Chapter 2

92

96

CHAPTER 3—FLUID FRICTION IN PIPES

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

Introduction

Laminar Flow

Example 3.1—Polymer Flow in a Pipeline

Models for Shear Stress

Piping and Pumping Problems

Example 3.2—Unloading Oil from a Tanker

Speciﬁed Flow Rate and Diameter

Example 3.3—Unloading Oil from a Tanker

Speciﬁed Diameter and Pressure Drop

Example 3.4—Unloading Oil from a Tanker

Speciﬁed Flow Rate and Pressure Drop

Example 3.5—Unloading Oil from a Tanker

Miscellaneous Additional Calculations

Flow in Noncircular Ducts

Example 3.6—Flow in an Irrigation Ditch

Compressible Gas Flow in Pipelines

Compressible Flow in Nozzles

Complex Piping Systems

Example 3.7—Solution of a Piping/Pumping Problem

Problems for Chapter 3

120

123

128

129

133

142

144

147

147

150

152

156

159

163

165

168

CHAPTER 4—FLOW IN CHEMICAL ENGINEERING EQUIPMENT

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

Introduction

Pumps and Compressors

Example 4.1—Pumps in Series and Parallel

Drag Force on Solid Particles in Fluids

Example 4.2—Manufacture of Lead Shot

Flow Through Packed Beds

Example 4.3—Pressure Drop in a Packed-Bed Reactor

Filtration

Fluidization

Dynamics of a Bubble-Cap Distillation Column

Cyclone Separators

Sedimentation

Dimensional Analysis

Example 4.4—Thickness of the Laminar Sublayer

Problems for Chapter 4

185

188

193

194

202

204

208

210

215

216

219

222

224

229

230

Contents

ix

PART II—MICROSCOPIC FLUID MECHANICS

CHAPTER 5—DIFFERENTIAL EQUATIONS OF FLUID MECHANICS

5.1

5.2

5.3

5.4

5.5

5.6

5.7

Introduction to Vector Analysis

Vector Operations

Example 5.1—The Gradient of a Scalar

Example 5.2—The Divergence of a Vector

Example 5.3—An Alternative to the Diﬀerential

Element

Example 5.4—The Curl of a Vector

Example 5.5—The Laplacian of a Scalar

Other Coordinate Systems

The Convective Derivative

Diﬀerential Mass Balance

Example 5.6—Physical Interpretation of the Net Rate

of Mass Outﬂow

Example 5.7—Alternative Derivation of the Continuity

Equation

Diﬀerential Momentum Balances

Newtonian Stress Components in Cartesian Coordinates

Example 5.8—Constant-Viscosity Momentum Balances

in Terms of Velocity Gradients

Example 5.9—Vector Form of Variable-Viscosity

Momentum Balance

Problems for Chapter 5

249

250

253

257

257

262

262

263

266

267

269

270

271

274

280

284

285

CHAPTER 6—SOLUTION OF VISCOUS-FLOW PROBLEMS

6.1

6.2

6.3

6.4

Introduction

Solution of the Equations of Motion in Rectangular

Coordinates

Example 6.1—Flow Between Parallel Plates

Alternative Solution Using a Shell Balance

Example 6.2—Shell Balance for Flow Between Parallel

Plates

Example 6.3—Film Flow on a Moving Substrate

Example 6.4—Transient Viscous Diﬀusion of

Momentum (COMSOL)

Poiseuille and Couette Flows in Polymer Processing

Example 6.5—The Single-Screw Extruder

Example 6.6—Flow Patterns in a Screw Extruder

(COMSOL)

292

294

294

301

301

303

307

312

313

318

x

Contents

6.5

6.6

Solution of the Equations of Motion in Cylindrical

Coordinates

Example 6.7—Flow Through an Annular Die

Example 6.8—Spinning a Polymeric Fiber

Solution of the Equations of Motion in Spherical

Coordinates

Example 6.9—Analysis of a Cone-and-Plate Rheometer

Problems for Chapter 6

322

322

325

327

328

333

CHAPTER 7—LAPLACE’S EQUATION, IRROTATIONAL AND

POROUS-MEDIA FLOWS

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

7.10

7.11

Introduction

Rotational and Irrotational Flows

Example 7.1—Forced and Free Vortices

Steady Two-Dimensional Irrotational Flow

Physical Interpretation of the Stream Function

Examples of Planar Irrotational Flow

Example 7.2—Stagnation Flow

Example 7.3—Combination of a Uniform Stream and

a Line Sink (C)

Example 7.4—Flow Patterns in a Lake (COMSOL)

Axially Symmetric Irrotational Flow

Uniform Streams and Point Sources

Doublets and Flow Past a Sphere

Single-Phase Flow in a Porous Medium

Example 7.5—Underground Flow of Water

Two-Phase Flow in Porous Media

Wave Motion in Deep Water

Problems for Chapter 7

354

356

359

361

364

366

369

371

373

378

380

384

387

388

390

396

400

CHAPTER 8—BOUNDARY-LAYER AND OTHER NEARLY

UNIDIRECTIONAL FLOWS

8.1

8.2

8.3

8.4

8.5

8.6

Introduction

Simpliﬁed Treatment of Laminar Flow Past a Flat Plate

Example 8.1—Flow in an Air Intake (C)

Simpliﬁcation of the Equations of Motion

Blasius Solution for Boundary-Layer Flow

Turbulent Boundary Layers

Example 8.2—Laminar and Turbulent Boundary

Layers Compared

Dimensional Analysis of the Boundary-Layer Problem

414

415

420

422

425

428

429

430

Contents

8.7

8.8

8.9

8.10

Boundary-Layer Separation

Example 8.3—Boundary-Layer Flow Between Parallel

Plates (COMSOL Library)

Example 8.4—Entrance Region for Laminar Flow

Between Flat Plates

The Lubrication Approximation

Example 8.5—Flow in a Lubricated Bearing (COMSOL)

Polymer Processing by Calendering

Example 8.6—Pressure Distribution in a Calendered

Sheet

Thin Films and Surface Tension

Problems for Chapter 8

xi

433

435

440

442

448

450

454

456

459

CHAPTER 9—TURBULENT FLOW

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

9.10

9.11

9.12

9.13

9.14

Introduction

Example 9.1—Numerical Illustration of a Reynolds

Stress Term

Physical Interpretation of the Reynolds Stresses

Mixing-Length Theory

Determination of Eddy Kinematic Viscosity and

Mixing Length

Velocity Proﬁles Based on Mixing-Length Theory

Example 9.2—Investigation of the von K´

arm´an

Hypothesis

The Universal Velocity Proﬁle for Smooth Pipes

Friction Factor in Terms of Reynolds Number for Smooth

Pipes

Example 9.3—Expression for the Mean Velocity

Thickness of the Laminar Sublayer

Velocity Proﬁles and Friction Factor for Rough Pipe

Blasius-Type Law and the Power-Law Velocity Proﬁle

A Correlation for the Reynolds Stresses

Computation of Turbulence by the k/ε Method

Example 9.4—Flow Through an Oriﬁce Plate (COMSOL)

Example 9.5—Turbulent Jet Flow (COMSOL)

Analogies Between Momentum and Heat Transfer

Example 9.6—Evaluation of the Momentum/HeatTransfer Analogies

Turbulent Jets

Problems for Chapter 9

473

479

480

481

484

486

487

488

490

491

492

494

495

496

499

501

505

509

511

513

521

xii

Contents

CHAPTER 10—BUBBLE MOTION, TWO-PHASE FLOW, AND

FLUIDIZATION

10.1

10.2

10.3

10.4

10.5

10.6

10.7

10.8

Introduction

Rise of Bubbles in Unconﬁned Liquids

Example 10.1—Rise Velocity of Single Bubbles

Pressure Drop and Void Fraction in Horizontal Pipes

Example 10.2—Two-Phase Flow in a Horizontal Pipe

Two-Phase Flow in Vertical Pipes

Example 10.3—Limits of Bubble Flow

Example 10.4—Performance of a Gas-Lift Pump

Example 10.5—Two-Phase Flow in a Vertical Pipe

Flooding

Introduction to Fluidization

Bubble Mechanics

Bubbles in Aggregatively Fluidized Beds

Example 10.6—Fluidized Bed with Reaction (C)

Problems for Chapter 10

531

531

536

536

541

543

546

550

553

555

559

561

566

572

575

CHAPTER 11—NON-NEWTONIAN FLUIDS

11.1

11.2

11.3

11.4

11.5

11.6

Introduction

Classiﬁcation of Non-Newtonian Fluids

Constitutive Equations for Inelastic Viscous Fluids

Example 11.1—Pipe Flow of a Power-Law Fluid

Example 11.2—Pipe Flow of a Bingham Plastic

Example 11.3—Non-Newtonian Flow in a Die

(COMSOL Library)

Constitutive Equations for Viscoelastic Fluids

Response to Oscillatory Shear

Characterization of the Rheological Properties of Fluids

Example 11.4—Proof of the Rabinowitsch Equation

Example 11.5—Working Equation for a CoaxialCylinder Rheometer: Newtonian Fluid

Problems for Chapter 11

591

592

595

600

604

606

613

620

623

624

628

630

CHAPTER 12—MICROFLUIDICS AND ELECTROKINETIC

FLOW EFFECTS

12.1

12.2

12.3

12.4

Introduction

Physics of Microscale Fluid Mechanics

Pressure-Driven Flow Through Microscale Tubes

Example 12.1—Calculation of Reynolds Numbers

Mixing, Transport, and Dispersion

639

640

641

641

642

Contents

12.5

12.6

12.7

12.8

12.9

Species, Energy, and Charge Transport

The Electrical Double Layer and Electrokinetic Phenomena

Example 12.2—Relative Magnitudes of Electroosmotic

and Pressure-Driven Flows

Example 12.3—Electroosmotic Flow Around a Particle

Example 12.4—Electroosmosis in a Microchannel

(COMSOL)

Example 12.5—Electroosmotic Switching in a

Branched Microchannel (COMSOL)

Measuring the Zeta Potential

Example 12.6—Magnitude of Typical Streaming

Potentials

Electroviscosity

Particle and Macromolecule Motion in Microﬂuidic Channels

Example 12.7—Gravitational and Magnetic Settling

of Assay Beads

Problems for Chapter 12

xiii

644

647

648

653

653

657

659

660

661

661

662

666

CHAPTER 13—AN INTRODUCTION TO COMPUTATIONAL

FLUID DYNAMICS AND FLOWLAB

13.1

13.2

13.3

13.4

Introduction and Motivation

Numerical Methods

Learning CFD by Using FlowLab

Practical CFD Examples

Example 13.1—Developing Flow in a Pipe

Entrance Region (FlowLab)

Example 13.2—Pipe Flow Through a Sudden

Expansion (FlowLab)

Example 13.3—A Two-Dimensional Mixing Junction

(FlowLab)

Example 13.4—Flow Over a Cylinder (FlowLab)

References for Chapter 13

671

673

682

686

687

690

692

696

702

CHAPTER 14—COMSOL (FEMLAB) MULTIPHYSICS FOR

SOLVING FLUID MECHANICS PROBLEMS

14.1

14.2

14.3

14.4

Introduction to COMSOL

How to Run COMSOL

Example 14.1—Flow in a Porous Medium with an

Obstruction (COMSOL)

Draw Mode

Solution and Related Modes

703

705

705

719

724

Contents

xiv

14.5

Fluid Mechanics Problems Solvable by COMSOL

Problems for Chapter 14

725

730

APPENDIX A:

USEFUL MATHEMATICAL RELATIONSHIPS

731

APPENDIX B:

ANSWERS TO THE TRUE/FALSE ASSERTIONS

737

APPENDIX C:

SOME VECTOR AND TENSOR OPERATIONS

740

INDEX

743

THE AUTHORS

753

PREFACE

T

HIS text has evolved from a need for a single volume that embraces a wide

range of topics in ﬂuid mechanics. The material consists of two parts—four

chapters on macroscopic or relatively large-scale phenomena, followed by ten chapters on microscopic or relatively small-scale phenomena. Throughout, I have tried

to keep in mind topics of industrial importance to the chemical engineer. The

scheme is summarized in the following list of chapters.

Part I—Macroscopic Fluid Mechanics

1. Introduction to Fluid Mechanics

2. Mass, Energy, and Momentum

Balances

3. Fluid Friction in Pipes

4. Flow in Chemical

Engineering Equipment

Part II—Microscopic Fluid Mechanics

5. Diﬀerential Equations of Fluid

Mechanics

6. Solution of Viscous-Flow Problems

7. Laplace’s Equation, Irrotational

and Porous-Media Flows

8. Boundary-Layer and Other

Nearly Unidirectional Flows

9. Turbulent Flow

10. Bubble Motion, Two-Phase Flow,

and Fluidization

11. Non-Newtonian Fluids

12. Microﬂuidics and

Electrokinetic Flow Eﬀects

13. An Introduction to

Computational Fluid

Dynamics and FlowLab

14. COMSOL (FEMLAB) Multiphysics for Solving Fluid

Mechanics Problems

In our experience, an undergraduate ﬂuid mechanics course can be based on

Part I plus selected parts of Part II, and a graduate course can be based on

much of Part II, supplemented perhaps by additional material on topics such as

approximate methods and stability.

Second edition. I have attempted to bring the book up to date by the major addition of Chapters 12, 13, and 14—one on microﬂuidics and two on CFD

(computational ﬂuid dynamics). The choice of software for the CFD presented

a diﬃculty; for various reasons, I selected FlowLab and COMSOL Multiphysics,

but there was no intention of “promoting” these in favor of other excellent CFD

programs.1 The use of CFD examples in the classroom really makes the subject

1

The software name “FEMLAB” was changed to “COMSOL Multiphysics” in September 2005, the ﬁrst

release under the new name being COMSOL 3.2.

xv

xvi

Preface

come “alive,” because the previous restrictive necessities of “nice” geometries and

constant physical properties, etc., can now be lifted. Chapter 9, on turbulence, has

also been extensively rewritten; here again, CFD allows us to venture beyond the

usual ﬂow in a pipe or between parallel plates and to investigate further practical

situations such as turbulent mixing and recirculating ﬂows.

Example problems. There is an average of about six completely worked examples in each chapter, including several involving COMSOL (dispersed throughout

Part II) and FlowLab (all in Chapter 13). The end of each example is marked by a

small, hollow square: . All the COMSOL examples have been run on a Macintosh

G4 computer using FEMLAB 3.1, but have also been checked on a PC; those using

a PC or other releases of COMSOL/FEMLAB may encounter slightly diﬀerent windows than those reproduced here. The format for each COMSOL example is: (a)

problem statement, (b) details of COMSOL implementation, and (c) results and

discussion (however, item (b) can easily be skipped for those interested only in the

results).

The numerous end-of-chapter problems have been classiﬁed roughly as easy

(E), moderate (M), or diﬃcult/lengthy (D). The University of Cambridge has given

permission—kindly endorsed by Professor J.F. Davidson, F.R.S.—for several of

their chemical engineering examination problems to be reproduced in original or

modiﬁed form, and these have been given the additional designation of “(C)”.

Acknowledgments.

I gratefully acknowledge the written contributions of

my former Michigan colleague Stacy Birmingham (non-Newtonian ﬂuids), Brian

Kirby of Cornell University (microﬂuidics), and Chi-Yang Cheng of Fluent, Inc.

(FlowLab). Although I wrote most of the COMSOL examples, I have had great help

and cooperation from COMSOL Inc. and the following personnel in particular—

Philip Byrne, Bjorn Sjodin, Ed Fontes, Peter Georen, Olof Hernell, Johan Linde,

and R´emi Magnard. At Fluent, Inc., Shane Moeykens was instrumental in identifying Chi-Yang Cheng as the person best suited to write the FlowLab chapter.

Courtney Esposito and Jordan Schmidt of The MathWorks kindly helped me with

MATLAB, needed for the earlier 2.3 version of FEMLAB.

I appreciate the assistance of several other friends and colleagues, including

Nitin Anturkar, Stuart Churchill, John Ellis, Kevin Ellwood, Scott Fogler, Leenaporn Jongpaiboonkit, Lisa Keyser, Kartic Khilar, Ronald Larson, Susan Montgomery, Donald Nicklin, the late Margaret Sansom, Michael Solomon, Sandra

Swisher, Rasin Tek, Robert Ziﬀ, and my wife Mary Ann Gibson Wilkes. Also very

helpful were Bernard Goodwin, Elizabeth Ryan, and Michelle Housley at Prentice Hall PTR, and my many students and friends at the University of Michigan

and Chulalongkorn University in Bangkok. Others are acknowledged in speciﬁc

literature citations.

Further information. The website http://www.engin.umich.edu/~fmche

is maintained as a “bulletin board” for giving additional information about the

Preface

xvii

book—hints for problem solutions, errata, how to contact the authors, etc.—as

proves desirable. My own Internet address is wilkes@umich.edu. The text was

composed on a Power Macintosh G4 computer using the TEXtures “typesetting”

program. Eleven-point type was used for the majority of the text. Most of the

ﬁgures were constructed using MacDraw Pro, Excel, and KaleidaGraph.

Professor Terence Fox , to whom this book is dedicated, was a Cambridge

engineering graduate who worked from 1933 to 1937 at Imperial Chemical Industries Ltd., Billingham, Yorkshire. Returning to Cambridge, he taught engineering

from 1937 to 1946 before being selected to lead the Department of Chemical Engineering at the University of Cambridge during its formative years after the end

of World War II. As a scholar and a gentleman, Fox was a shy but exceptionally

brilliant person who had great insight into what was important and who quickly

brought the department to a preeminent position. He succeeded in combining an

industrial perspective with intellectual rigor. Fox relinquished the leadership of

the department in 1959, after he had secured a permanent new building for it

(carefully designed in part by himself).

Fox was instrumental in bringing Peter Danckwerts, Kenneth Denbigh, John Davidson, and others into the department. He also accepted me in 1956 as a junior faculty member, and I spent four good years in the Cambridge University Department of Chemical Engineering.

Danckwerts subsequently wrote an appreciation2 of Fox’s

talents, saying, with almost complete accuracy: “Fox instigated no research and published nothing.” How times

have changed—today, unless he were known personally,

his r´esum´e would probably be cast aside and he would

stand little chance of being hired, let alone of receiving tenure! However, his lectures, meticulously written handouts, enthusiasm, genius, and friendship were

T.R.C. Fox

a great inspiration to me, and I have much pleasure in

acknowledging his positive impact on my career.

James O. Wilkes

August 18, 2005

2

P.V. Danckwerts, “Chemical engineering comes to Cambridge,” The Cambridge Review , pp. 53–55, February 28, 1983.

This page intentionally left blank

Some Greek Letters

α

β

γ, Γ

δ, Δ

,ε

ζ

η

θ, ϑ, Θ

ι

κ

λ, Λ

μ

alpha

beta

gamma

delta

epsilon

zeta

eta

theta

iota

kappa

lambda

mu

ν

ξ, Ξ

o

π, , Π

ρ,

σ, ς, Σ

τ

υ, Υ

φ, ϕ, Φ

χ

ψ, Ψ

ω, Ω

nu

xi

omicron

pi

rho

sigma

tau

upsilon

phi

chi

psi

omega

Chapter 1

INTRODUCTION TO FLUID MECHANICS

1.1 Fluid Mechanics in Chemical Engineering

A

knowledge of ﬂuid mechanics is essential for the chemical engineer because

the majority of chemical-processing operations are conducted either partly or

totally in the ﬂuid phase. Examples of such operations abound in the biochemical,

chemical, energy, fermentation, materials, mining, petroleum, pharmaceuticals,

polymer, and waste-processing industries.

There are two principal reasons for placing such an emphasis on ﬂuids. First,

at typical operating conditions, an enormous number of materials normally exist

as gases or liquids, or can be transformed into such phases. Second, it is usually

more eﬃcient and cost-eﬀective to work with ﬂuids in contrast to solids. Even

some operations with solids can be conducted in a quasi-ﬂuidlike manner; examples are the ﬂuidized-bed catalytic reﬁning of hydrocarbons, and the long-distance

pipelining of coal particles using water as the agitating and transporting medium.

Although there is inevitably a signiﬁcant amount of theoretical development,

almost all the material in this book has some application to chemical processing

and other important practical situations. Throughout, we shall endeavor to present

an understanding of the physical behavior involved; only then is it really possible

to comprehend the accompanying theory and equations.

1.2 General Concepts of a Fluid

We must begin by responding to the question, “What is a ﬂuid?” Broadly

speaking, a ﬂuid is a substance that will deform continuously when it is subjected

to a tangential or shear force, much as a similar type of force is exerted when

a water-skier skims over the surface of a lake or butter is spread on a slice of

bread. The rate at which the ﬂuid deforms continuously depends not only on the

magnitude of the applied force but also on a property of the ﬂuid called its viscosity

or resistance to deformation and ﬂow. Solids will also deform when sheared, but

a position of equilibrium is soon reached in which elastic forces induced by the

deformation of the solid exactly counterbalance the applied shear force, and further

deformation ceases.

3

4

Chapter 1—Introduction to Fluid Mechanics

A simple apparatus for shearing a ﬂuid is shown in Fig. 1.1. The ﬂuid is

contained between two concentric cylinders; the outer cylinder is stationary, and

the inner one (of radius R) is rotated steadily with an angular velocity ω. This

shearing motion of a ﬂuid can continue indeﬁnitely, provided that a source of

energy—supplied by means of a torque here—is available for rotating the inner

cylinder. The diagram also shows the resulting velocity proﬁle; note that the

velocity in the direction of rotation varies from the peripheral velocity Rω of the

inner cylinder down to zero at the outer stationary cylinder, these representing

typical no-slip conditions at both locations. However, if the intervening space

is ﬁlled with a solid—even one with obvious elasticity, such as rubber—only a

limited rotation will be possible before a position of equilibrium is reached, unless,

of course, the torque is so high that slip occurs between the rubber and the cylinder.

Fixed

cylinder

Rotating

cylinder

Rotating

cylinder

A

Velocity

profile

A

R

ω

Fluid

Fixed

cylinder

(a) Side elevation

Fluid

Rω

(b) Plan of section across A-A (not to scale)

Fig. 1.1 Shearing of a ﬂuid.

There are various classes of ﬂuids. Those that behave according to nice and obvious simple laws, such as water, oil, and air, are generally called Newtonian ﬂuids.

These ﬂuids exhibit constant viscosity but, under typical processing conditions,

virtually no elasticity. Fortunately, a very large number of ﬂuids of interest to the

chemical engineer exhibit Newtonian behavior, which will be assumed throughout

the book, except in Chapter 11, which is devoted to the study of non-Newtonian

ﬂuids.

A ﬂuid whose viscosity is not constant (but depends, for example, on the

intensity to which it is being sheared), or which exhibits signiﬁcant elasticity, is

termed non-Newtonian. For example, several polymeric materials subject to deformation can “remember” their recent molecular conﬁgurations, and in attempting

to recover their recent states, they will exhibit elasticity in addition to viscosity.

Other ﬂuids, such as drilling mud and toothpaste, behave essentially as solids and

1.3—Stresses, Pressure, Velocity, and the Basic Laws

5

will not ﬂow when subject to small shear forces, but will ﬂow readily under the

inﬂuence of high shear forces.

Fluids can also be broadly classiﬁed into two main categories—liquids and

gases. Liquids are characterized by relatively high densities and viscosities, with

molecules close together; their volumes tend to remain constant, roughly independent of pressure, temperature, or the size of the vessels containing them. Gases,

on the other hand, have relatively low densities and viscosities, with molecules

far apart; generally, they will rapidly tend to ﬁll the container in which they are

placed. However, these two states—liquid and gaseous—represent but the two

extreme ends of a continuous spectrum of possibilities.

P

Vaporpressure

curve

•C

L•

G•

T

Fig. 1.2 When does a liquid become a gas?

The situation is readily illustrated by considering a ﬂuid that is initially a gas

at point G on the pressure/temperature diagram shown in Fig. 1.2. By increasing

the pressure, and perhaps lowering the temperature, the vapor-pressure curve is

soon reached and crossed, and the ﬂuid condenses and apparently becomes a liquid

at point L. By continuously adjusting the pressure and temperature so that the

clockwise path is followed, and circumnavigating the critical point C in the process,

the ﬂuid is returned to G, where it is presumably once more a gas. But where does

the transition from liquid at L to gas at G occur? The answer is at no single point,

but rather that the change is a continuous and gradual one, through a whole

spectrum of intermediate states.

1.3 Stresses, Pressure, Velocity, and the Basic Laws

Stresses. The concept of a force should be readily apparent. In ﬂuid mechanics, a force per unit area, called a stress, is usually found to be a more convenient

and versatile quantity than the force itself. Further, when considering a speciﬁc

surface, there are two types of stresses that are particularly important.

1. The ﬁrst type of stress, shown in Fig. 1.3(a), acts perpendicularly to the

surface and is therefore called a normal stress; it will be tensile or compressive,

depending on whether it tends to stretch or to compress the ﬂuid on which it acts.

The normal stress equals F/A, where F is the normal force and A is the area of

the surface on which it acts. The dotted outlines show the volume changes caused

6

Chapter 1—Introduction to Fluid Mechanics

by deformation. In ﬂuid mechanics, pressure is usually the most important type

of compressive stress, and will shortly be discussed in more detail.

2. The second type of stress, shown in Fig. 1.3(b), acts tangentially to the

surface; it is called a shear stress τ , and equals F/A, where F is the tangential

force and A is the area on which it acts. Shear stress is transmitted through a

ﬂuid by interaction of the molecules with one another. A knowledge of the shear

stress is very important when studying the ﬂow of viscous Newtonian ﬂuids. For

a given rate of deformation, measured by the time derivative dγ/dt of the small

angle of deformation γ, the shear stress τ is directly proportional to the viscosity

of the ﬂuid (see Fig. 1.3(b)).

F

F

Area A

F

F

Fig. 1.3(a) Tensile and compressive normal stresses F/A, acting on a cylinder, causing elongation and shrinkage, respectively.

F

Original

position

Deformed

position

γ

Area A

F

Fig. 1.3(b) Shear stress τ = F/A, acting on a rectangular

parallelepiped, shown in cross section, causing a deformation

measured by the angle γ (whose magnitude is exaggerated here).

Pressure. In virtually all hydrostatic situations—those involving ﬂuids at

rest—the ﬂuid molecules are in a state of compression. For example, for the

swimming pool whose cross section is depicted in Fig. 1.4, this compression at a

typical point P is caused by the downwards gravitational weight of the water above

point P. The degree of compression is measured by a scalar, p—the pressure.

A small inﬂated spherical balloon pulled down from the surface and tethered

at the bottom by a weight will still retain its spherical shape (apart from a small

distortion at the point of the tether), but will be diminished in size, as in Fig.

1.4(a). It is apparent that there must be forces acting normally inward on the

1.3—Stresses, Pressure, Velocity, and the Basic Laws

7

surface of the balloon, and that these must essentially be uniform for the shape to

remain spherical, as in Fig. 1.4(b).

Water

Balloon

Surface

Balloon

Water

•

P

(b)

(a)

Fig. 1.4 (a) Balloon submerged in a swimming pool; (b) enlarged

view of the compressed balloon, with pressure forces acting on it.

Although the pressure p is a scalar, it typically appears in tandem with an area

A (assumed small enough so that the pressure is uniform over it). By deﬁnition

of pressure, the surface experiences a normal compressive force F = pA. Thus,

pressure has units of a force per unit area—the same as a stress.

The value of the pressure at a point is independent of the orientation of any

area associated with it, as can be deduced with reference to a diﬀerentially small

wedge-shaped element of the ﬂuid, shown in Fig. 1.5.

π −

θ

2

z

pA dA

dA

dC

θ

y

pC dC

dB

x

p B dB

Fig. 1.5 Equilibrium of a wedge of ﬂuid.

Due to the pressure there are three forces, pA dA, pB dB, and pC dC, that act

on the three rectangular faces of areas dA, dB, and dC. Since the wedge is not

moving, equate the two forces acting on it in the horizontal or x direction, noting

that pA dA must be resolved through an angle (π/2 − θ) by multiplying it by

cos(π/2 − θ) = sin θ:

pA dA sin θ = pC dC.

(1.1)

The vertical force pB dB acting on the bottom surface is omitted from Eqn. (1.1)

because it has no component in the x direction. The horizontal pressure forces

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